Although there are many results of hydrodynamic limits from the classical Boltzmann equation, the research on fluid limits from quantum Boltzmann equation is much limited. The compressible Euler and Navier-Stokes limits and incompressible Navier-Stokes limits from BFD equation for both hard potential and soft potential are formally derived by Zakrevskiy [1]}. We also mention that, Filbet, Hu and Jin [2]} introduced a new scheme for quantum Boltzmann equation to capture the Euler limit by numerical computations. However, due to the absence of rigorous mathematical theory on fluid limits from quantum Boltzmann equation, an attempt on rigorous proof of incompressible Navier-Stokes-Fourier limit from BFD equation is made by Jiang-Xiong-Zhou [3]} recently. In [3]} , the authors prove the global existence of classical solutions near equilibrium and in addition, they also obtain the uniform in \(\varepsilon \)  energy estimates, by which and the structure of equations of conservation laws they establish the incompressible Navier-Stokes-Fourier limit from BFD equation.
